Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix – Basic Definitions Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix – Properties Matrices A, B and C with elements a ij, b ij and c ij, respectively. 1. Equality For A and B each be m by n arrays Matrix A = Matrix B if and only if a ij = b ij for all values of i and j. 2. Addition A + B = C if and only if a ij + b ij = c ij for all values of i and j. For A, B and C each be m by n arrays 3. Commutative A + B = B + A 4. Associative (A + B) + C = A + (B + C) If B = O (the null matrix), for all A : A + O = O + A = A 5. Multiplication (by a Scalar) αA = (α A)in which the elements of αA are α a ij Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Multiplication, Inner Product if and only if Matrix multiplication * In general, matrix multiplication is not commutative ! commutator bracket symbol But if A and B are each diagonal * associative * distributive The product theorem For two n × n matrices A and B Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Multiplication, Inner Product Successive multiplication of row i of A with column j of B – row by column multiplication For example : [2 × 3] × [3 × 2] = [2 × 2] Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Multiplication, Inner Product For example : [3 × 2] × [2 × 2] = [3 × 2] Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Unit Matrix, Null Matrix The unit matrix 1 has elements δ ij, Kronecker delta, and the property that 1A = A1 = A for all A The null matrix O has all elements being zero ! Exercise 3.2.6(a) : if AB = 0, at least one of the matrices must have a zero determinant. If A is an n × n matrix with determinant  0, then it has a unique inverse A -1 so that AA -1 = A -1 A = 1. Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Direct product --- The direct tensor or Kronecker product If A is an m × m matrix and B an n × n matrix The direct product C is an mn × mn matrix with elements with For instance, if A and B are both 2 × 2 matrices The direct product is associative but not commutative ! Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Diagonal Matrices If a 3 × 3 square matrix A is diagonalIn any square matrix the sum of the diagonal elements is called the trace. 1. The trace is a linear operation : 2. The trace of a product of two matrices A and B is independent of the order of multiplication : (even though AB  BA) 3. The trace is invariant under cyclic permutation of the matrices in a product. Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Inversion Matrix A An operator that linearly transforms the coordinate axes Matrix A -1 An operator that linearly restore the original coordinate axes The elements Where C ji is the jith cofactor of A. For example : The cofactor matrix C and Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Inversion For example : |A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2 The elements of the cofactor matrix are Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Special matrices A matrix is called symmetric if: A T = A A skew-symmetric (antisymmetric) matrix is one for which: A T = -A An orthogonal matrix is one whose transpose is also its inverse: A T = A -1 Any matrix symmetricantisymmetric Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Inverse Matrix, A -1 The reverse of the rotation Transpose Matrix, Defining a new matrix such that holds only for orthogonal matrices ! Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Eigenvectors and Eigenvalues A is a matrix, v is an eigenvector of the matrix and λ the corresponding eigenvalue. This only has none trivial solutions for det (A- λ I) = 0. This gives rise to the secular equation for the eigenvalues: Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Eigenvectors and Eigenvalues Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Eigenvectors and Eigenvalues Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Example Eigenvalues and Eigenvectors of a real symmetric matrix The secular equation λ = -1,0,1 λ = -1.  x+y = 0, z = 0 Normalized λ = 0  x = 0, y = 0 λ = 1  -x+y = 0, z = 0 Normalized Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Example Degenerate Eigenvalues The secular equation λ = -1,1,1 λ = -1.  2x = 0, y+z = 0 Normalized λ = 1  -y+z = 0 (r 1 perpendicular to r 2 ) λ = 1  Normalized (r 3 must be perpendicular to r 1 and may be made perpendicular to r 2 ) Chapter 3 Systems of Differential Equations

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Conversion of an nth order differential equation to a system of n first-order differential equations Setting,,, …… ……

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Example : Mass on a spring assume eigenvector

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Homogeneous systems with constant coefficients in components y 1 y 2 -plane is called the phase plane Critical point : the point P at which dy 2 /dy 1 becomes undetermined is called P : (y 1,y 2 ) = (0,0)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Five Types of Critical points

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Criteria for Types of Critical points P is the sum of the eigenvalues, q the product and  the discriminant.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Stability Criteria for Critical points

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Example : Mass on a spring p = -c/m, q = k/m and  = (c/m) 2 -4k/m No damping c = 0 : p = 0, q > 0  a center Underdamping c 2 0,  < 0  a stable and attractive spiral point. Critical damping c 2 = 4mk : p 0,  = 0  a stable and attractive node. Overdamping c 2 > 4mk : p 0,  > 0  a stable and attractive node.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations No basis of eigenvectors available. Degenerate node If matrix A has a double eigenvalue  since If matrix A has a triple eigenvalue 

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations No basis of eigenvectors available. Degenerate node